Cauchy Problem for Hyper-Parabolic Partial Differential Equations

Abstract

This chapter focuses on the Cauchy problem for hyper-parabolic partial differential equations. The chapter considers initial-boundary-value problems for partial differential equations of the form u t = u xx – u yy . Such equations arise in a discussion of classification. Specifically, every linear second-order constant-coefficient partial differential equation in n variable can be reduced to a specific form discussed in the chapter. Equations of this type have arisen in diverse nonstandard applications, most of which require a solution subject to classical initial and boundary conditions on a space-time cylinder. However such a problem is not well-posed but is “hyper-sensitive” to variations in the data. The chapter develops some elementary notions of generalized solutions of an abstract model as an evolution equation in Hilbert space. The chapter then presents some well-posed problems for this equation, and these results suggest a more natural method of approximating solutions of the ill-posed Cauchy problem.